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Appendix II.                             857

And, slope ordinate = (stress sum-curve ordinate) —


Thus the slope curve s may be drawn by summating the stress
curve, and then dividing by Ey.

Again, if A B be made very small, the angle a = VA B, and tan a
= average slope between A B. Then,

Deflection at B = D tana = D (average slope A to B).
Similarly  the   angle  (a + /3) = inclination  of  B c   to   A v,   and
tan (a + /3) = average slope between B c.    Then,

Total deflection at c = D tan a -I- F tan (a -h/3)
= D (slope A B) -f F (slope B c) = ef+gh

And, sum of slope curve 1 __ f Total deflection
to any point           J """ \   at that point.

The problem is therefore completed by drawing the deflection
curve A as the summation of slope curve area, and the general
formula is deduced for a beam of uniform section :

Deflection - *<«=) _ S(SBJ
ZEy           El

To prove that the maximum deflection of a cantilever with
concentrated load = W/3/3El (p.451): the Bm curve is triangular,
whose sum = W/2/2, and this is the maximum slope. But the
slope curve is a parabola, whose area is therefore |- W/2/2x/
viz., W /3/3, which is the second summation of the Bm; and by
above general formula, deflection = W /3/3 E I.

Some care is required in fixing scales, but previous explana-
tions may be consulted. When beam section varies, Z will alter
also, and the stress curve must be found : then continue as before.
Note that load^ shear, Bm , slope^ and deflection curves are a con-
tinuous series, where each is the sum or integral of the preceding

P. 458. Pillars and Struts.—In the paper cited on
p. 458, Prof. Fidler assigns various reasons why pillar strength
cannot be shewn practically by Euler's formula, such as an in-
constant E, even in the same strut, and initial curvature in line of
thrust, the latter altering W considerably. In Fig. 820 the crosses
shew Christie's experiments on T bars, and the small circles